The is some logic to this order which does present the proofs in a more structured way than if presented say in the number order of the multiplier.

Multiplying by Eleven

The rule for multiplying by eleven is to “add the neighbor”. The neighbor being the digit to the right of the digit we are looking at.

We will write a zero at the front of the number and we apply the “add the neighbor” rule to the zero as well. The digit on the far right of the number has no neighbor so nothing is added to this digit.

We start with our general number as a four digit number.

We will multiply this general number by 11 but we will do so by separating 11 into 10 plus 1.

We will multiply by 10 then multiply it by 1 and add the them together. When multiplying by 10 we will also add a zero to our general number.

Which is the rule for multiplying by eleven. You can see each letter is added to its neighbor, in the case of it has no neighbor, so is added to 0, then is added to , is added to , is added to then is added to 0.

Multiplying by Twelve

The rule for multiplying by twelve is to “Double and add the neighbor” with the neighbor being the digit to the right of the digit we are looking at.
We start with our general number as a four digit number.

We will multiply this general number by 12 but we will do so by separating 12 into 10 plus 2.

We will multiply by 10 then multiply it by 2 and add the them together. When multiplying by 10 we will also add a zero to our general number.

We now multiply by 2, which will double the value.

Adding the two together.

Grouping the different units together we get.

The Commutative Property of Addition states that changing the order of addends does not change the sum so we can swap the order of the addends in each of the parentheses.

Which is the rule for multiplying by twelve. You can see each letter is doubled and added to its neighbor, in the case of it has no neighbor, so is added to 0, then is added to , is added to , is added to then is added to 0.

Multiplying by Six

We will look at the rule for multiplying by six next as the rule is similar to that of eleven but we only add half the neighbor and add five if it is odd, meaning the number itself is odd not if the neighbor is odd.
Six is five plus one and we can think of five as being ten times one half.

We start with our general number as a four digit number.

We will multiply by 6

Grouping the different units together we get.

We can add a zero to the first term and write in the form

This is the rule for six where you add half the neighbor to the number which is fine if all the digits were always even numbers. However, the rule says "to add half the neighbor and add five if the number is odd."
We will assume that is odd and so replace it with where is what we have called the "smaller half" of an odd number.
For example, and

We replace the with and we have the rest of the rule for six. When is odd we add 5 and for the next digit we add which is the "smaller half" of .

Multiplying by Seven

We will look at the rule for multiplying by seven next as the rule is similar to that of six but we double the number.
Seven is five plus two and we can think of five as being ten times one half.

We start with our general number as a four digit number.

We will multiply by 7

Grouping the different units together we get.

We can add a zero to the first term and write in the form

This is the rule for seven where you double the number and add half the neighbor which is fine if all the digits were always even numbers. However, the rule says "to add half the neighbor and add five if the number is odd."
We will again assume that is odd and so replace it with where is what we have called the "smaller half" of an odd number.

We replace the with and we have the rest of the rule for seven. When is odd we add 5 and for the next digit we add which is the "smaller half" of .

Multiplying by Nine

The rules for multiplying by nine are:
1 - Subtract the right hand figure from 10
2 - Subtract each other figure from 9 and add the neighbor.
3 - On the left hand digit, use the left hand digit minus 1.

We will start with the fact that 9 is .

We can add and subtract any number to our general number without changing the value, for example is still . The benefit of this step is it will allow us to group the added and subtracted values which will allow us to simplify the groups of terms. We will add and subtract 9000, 900, 90 and 9 to each of their respective terms

We can now group together the terms for each unit.

We can replace the with

And there we have the rule for multiplying by nine.
We can generalize this rule to numbers of any length as:

Multiplying by Four

The rules for multiplying by four are:
1 - Subtract the right hand figure from 10, add five if odd.
2 - Subtract each other figure from 9, add five if odd and add half the neighbor.
3 - at the zero in front of the number use half the left hand digit minus 1.

We will start with the fact that 4 is and 5 is .

We can add and subtract 9000, 900, 90 and 9 to each of their respective terms as we have done in other proofs.

We can now group together the terms for each unit.

We can replace the with

And there we have the rule for multiplying by four, as long as all the digits are even. However, the rule says "to add half the neighbor and add five if the number is odd."
We will again assume that is odd and so replace it with where is what we have called the "smaller half" of an odd number.

We replace the with and we have the rest of the rule for four. When is odd we add 5 and for the next digit we add which is the "smaller half" of .

Multiplying by Eight

The rule for multiplying by eight is:
1 - Subtract the right hand figure from 10 and double
2 - Subtract each other figure from 9, double and add the neighbor.
3 - On the left hand digit, use the left hand digit minus 2.

For 8 will will start with the fact that 8 is .

Like we did for nine we will now add and subtract a number but instead of 9000 etc, since we are doubling for eight we will use 18000, 1800, 180 and 18 this time.

We can now group together the terms for each unit.

We can replace the with .

And there we have the rule for multiplying by eight.
We can generalize this rule to numbers of any length as:

Multiplying by Three

The rules for multiplying by three are:
1 - Subtract the right hand figure from 10, double and add 5 if odd.
2 - Subtract each other figure from 9, double and add half the neighbor and add 5 if odd.
3 - On the left hand digit, use half the left hand digit minus 2.

The rule for three is very similar to the rule for eight and so we will follow a similar logic to that we followed for eight.
For three we will start with the fact that 3 is .

As we did for eight we will now add and subtract 18000, 1800, 180 and 18.

We can group together the same units.

We can replace the with

And there we have the rule for multiplying by three, if all the digits were always even. However, the rule says "to add half the neighbor and add five if the number is odd."
As we did earlier for six we will assume that is odd and so replace it with .

We replace the with and we have the rest of the rule for three. When is odd we add 5 and for the next digit we add which is the "smaller half" of .